 We have discussed the first law and the second law in class, I am going to discuss some illustrative problems. The first problem I want to discuss is about historical measurements of the mechanical equivalent of heat, the experiments that were done by Rumpford, Mayer and Joule. The numbers I am going to give you are equivalent numbers and they are only illustrative and do not reflect the care with which original measurements were made. There is a book called The World of Physics by Weaver, Shyamalan's sister in 1987, I would recommend that you look at it for details. In each of the cases what I want to do is to calculate the mechanical equivalent of heat. So the first experiment was by Count Rumpford in his Canon boring experiment as it was called. He demonstrated that the horse working for two and a half hours could generate enough heat by friction to bring 12 kilograms of water at 0 degrees to its boiling point. The horse power is a vague unit, you can take one horse power to be equal to 0.75 kilowatts which is the current definition of horse power. The experiment was with you to Mayer who showed that the falling of a mass through 365 meters can raise the temperature of an equal mass of water through 1 degree C. The third experiment consists of the famous paddle wheel experiments by Joule between 1840 and 1815 who demonstrated that on an average 2.32 watt hours of work produced a temperature rise of 0.56 degrees in 3.5 kgs of water. Joule also showed that using 12 volts across a coil fully immersed in water and passing 3 amps of current for 5 minutes raise the temperature of 180 grams of water through 14.3 degrees. So in each of these cases we will calculate J the mechanical equivalent. Take the experiment of Count Trumford, we are given that one horse power is 0.75 kilowatts. Now I am just calculating the work done by a horse in two and a half hours. This is one number of horses, 2.5 is ours, 0.75 is kilowatt by horse power then this is seconds per hour. So if you multiply these out you get kilowatt seconds kilojoules you get 6, 7, 5, 0 kilojoules. Now the heat generated as a consequence was given by in the experiment it is given to you it is given as 12 kilograms times 1 kilo calorie per kilogram degree C and this is degree C. So you get simply 1200 kilo calories J is simply W by Q. So if you divide this 6, 7, 5, 0 kilojoules by 1200 you get 5.625. You can see this is a bit high but Count Trumford himself used a little lower value than 0.75 is the conversion factor from what is now called the kilowatt unit to the horse power. In any case he got a number conceptually this is one of the first demonstrations of the equivalence of heat and work. Now Mayer did this experiment again he reported that a given mass falling through 365 meters so you have mass let us say in kilograms this is meters per second squared. So you get kilogram into meters per second square Newtons and this is meters. So Newton meter being a joule he gets a essentially 3580 M joules as the work done the corresponding heat generated is given by again the mass is in kilograms. This is kilo calories per kilogram degree C this is degree C this is 1000 is the conversion from calorie kilo calories to calories. So this gives you 1000 M because he said an equivalent mass of water. So J according to Mayer was simply W by Q which was 3581 by 1000 came to 3.58 this is lower than the current value that is accepted it is about 4.18. Then there were the experiments by Jules some of Jules paddle wheel experiments are the most famous you may recall that he actually did this experiment repeatedly in the Royal Society till the members of the Royal Society actually protested and said we are convinced do not do any more experiments but Jules insisted that he do experiments in various mean ways. The paddle wheel experiments essentially consisted of having a paddle wheel, a container and a paddle wheel this paddle wheel was rotated through a system a complicated system of pulleys and ropes etc and a weight finally this weight could move up and down and you must remember that Jules did his experiments with great care he accounted for convection loss radiation loss to the environment accounted for the elasticity of the rope he accounted for the kinetic energy of the weight sitting the ground and so on and that world of physics reference that I told you about these are given in detail. And this whole thing was an arrangement of pulleys and ropes okay now let us get to his experiment as experiment said that the total work done was 2.32 watt hours this is watt hours this is seconds per hour so if you convert it into watt seconds which is Jules which will give you 8352 Jules found that 3.565 which is kgs of water this is kilo calories per kg per degree C this is degree C and I am multiplying by 1000 in order to get calories. So you get 196.4 calories is equivalent to 83.52 Jules or the mechanical equivalent of heat is 4.184 is as close to the currently accepted value as you can think of now let us look at another experiment that Jules did but I must emphasize here that you really have to read the original experiments in the care with which Jules did these experiments because Jules was the first one to assert that while it was known that work and heat where the only two ways in which a closed system can exchange energy sorry while it was known that work and heat where two ways of exchanging energy with the surroundings for a closed system Jules is the one who first asserted that there are only two ways to make that assertion he needed to show that work done in any form was equivalent to the heat and everything every form of exchange of energy was actually either heat or work. Now let us look at another experiment that Jules did this is an experiment in electric current Jules had 12 volts he had 3 amperes passing through the circuit he had essentially a container with water well stirred maybe more importantly he passed a current through it so he passes 3 amperes through a 12 volt circuit so you get this and this is for five minutes so this is seconds so you get essentially what seconds which gives you Jules so it is 10,800 Jules was the work done by the current passing through the circuit for five minutes. Now the corresponding increase in water temperature is given and the weight of the water is given it is 180 kilograms this is kilo calories per degree C per kilogram and this is degree C so what you get by way of Q is simply a total of I am sorry the experiment was simply grams not kilograms so here you are talking about calories per gram in the total amount here you get this 2574 calories and J is therefore 10,800 divided by 2.574 you get 4.19. Again you have to read the original description the original experiments to appreciate the accuracy the methodical way in which Jules did these experiments when you are a pioneer it becomes that much more difficult because you have to be extra careful it shows a certain perseverance and faith in the scientific process then let me now get to a second problem we can discuss both closed and open systems in class and you can calculate the work done in different situations and here we are going to look at a pump that works adiabatically this is a pump at the inlet this is 1 kg per centimeter squared and the outlet is given as 10 kg per centimeter squared and the flow rate is given as 10 liters per second you should realize that the pump itself if you look at it in cross section is driven by a motor so this is the input to the pump this is the outlet and this is the motor working on the pump so question is how much horsepower should the motor have in order to do this job horsepower or kilowatts what should be the power of the motor so it says the pump works adiabatically it is required to pump water at 25 degrees and 1 kg per centimeter squared to 10 kg per centimeter squared at the rate of 10 liters per second at constant entropy specific volume varies with pressure this comes from painstaking measurements of the equation of state you get an expression V is equal to 10 25 – 0.5076 P where V is in centimeters per kilogram centimeter cube per kilogram and P is in kg per centimeter squared so you are asked to calculate the rating of the motor pump motor driving the pump in kilowatts so we have to define a system we define the system as simply the contents of the pump here we are saying the contents of the pump you essentially analyze the system for a time long enough so that this is less than total water flowing through it is a 10 liters per second so you are talking of even if you are talking of 10 seconds you are talking of 100 liters going through and the size of the pump is much smaller so what the pump contains is small mass which is small mass of water and the mass can be neglected in comparison to the mass of water flowing through during the period for which the analysis is done or you can discuss the operation of the pump at steady state so it is either steady state or negligible hold up hold up is what is contained inside the pump when you can write the two laws we have written this in class for an open system it says adiabatic so this has to be 0 you are interested in calculating this because of the negligible hold up or the steady state assumption this is 0 so you simply have at steady state three terms of which dm in is equal to dm out so if you take ws dot by m dot the rate at which work is done divided by the rate at which mass is flowing through the system you simply get minus of delta H you take these two to the other side is h out minus h in so ws equal to plus sorry this is defined as h out minus h in you can calculate h out minus h in from again the equations of thermodynamics you have dh is equal to delta Q plus V dp looking at PVT systems and here delta Q is given as 0 so dh delta h is simply the integral over V dp from P in to P out which is from 1 to 10 some conversion involved because this expression that is given to you in the problem statement gives you the volume in centimeter cube per kilogram the pressure itself is given in kilograms per centimeter squared so you get this is kilogram into centimeter these kilograms sorry kilogram and kilogram cancel and when you do the V dp calculation this is kilogram per centimeter squared pressure this is the pressure should be multiplied by 9.81 which gives you meters per second squared so this is the actual pressure kilogram meter per second squared will give you new tens it will give you new tens per centimeter squared so essentially this centimeter is converted to meter here this is meters per centimeter and finally you are dividing by thousand in order to get kilo joules out of this you get joules and this is the conversion from joules to kilo joules so the result you get is simply 0.97 kilo joules per kilogram now this is liters per second this is the density which is kg per liter of water into the work done per unit mass which is simply 0.97 so you get 9.7 kilowatt so this is the rating of the pump that you need for adiabatic operation to pump 10 liters per second of water from 1 kg to 10 kg per centimeter squared I would recommend that you get an idea of the order of magnitude of this by looking at the rating of the pump maybe in your house where the water is pumped from the tank underground to the overhead tank so this is a second example where we have we calculate the work done by the pump and we have assumed adiabatic effectively during the process of pumping the time is rather short for much heat to be exchanged the surroundings sets the temperature difference is also small between the pump or its contents and the surroundings it is a very reasonable assumption so you get the rating of the pump directly this is W s dot like you get W s dot here which is the power let me take a third example all of our big cities are short of power and one suggestion is that you could set up a power plant on the banks of a river in this case we are talking of Chennai I am looking at the river Cuam in order to set up a power plant because Karnau has told you that if you start with energy at temperature T1 you need a temperature T2 to which you have to reject heat you draw heat at T1 reject heat at T2 and you can get work at most equal to T1 minus T2 by T1 into the amount of heat drawn now the amount of heat drawn is one constraint another constraint could be the amount of heat rejected so if you have for example let us say you have a river flowing you have a nuclear reactor so you are drawing heat from it you are rejecting heat here and you are doing work here right this W is related to this amount of heat rejected here or W dot is rejected is related to Q2 now what you are asked is because of pollution considerations you are told that there is this river here Cuam river in this case in which the flow varies seasonally and in all seasons you are allowed to pollute the river thermally to the extent of 1 degree centigrade for example in the dry season it runs at 30 degrees C at 2 meter cube per second so you can go from 30 to 31 degrees beyond that you will affect the life in the river similarly in the wet season for example the water is at 25 degrees and the flow is at 8 meter cube per second again you are allowed to pollute it to the extent of 1 degree you are asked to calculate what is the maximum power you go back to Karno Karno tells you that this reactor here in this case is at 400 degrees C and this temperature varies with the season here this is T2 so the reactor temperature is T1 it is equal to 673 degrees K Kelvin now W dot by Q2 dot is if you like Q1 dot minus Q2 dot by Q1 dot but this is equal to T1 minus T2 by T2 it is what Karno has told now T2 vary with season so you are asked to calculate in two seasons what is the amount of work that amount of work that power that you can produce the dry season flow rate is given as 2 meter cube per second and the temperature you already been told 30 degrees which is 303.2 degrees K so you can use this formula you know W dot by Q dot is simply Q1 minus Q2 is W dot divided by Q2 which is T1 minus T2 by T2 so if Q2 dot permitted is now given to you it is given as 2 into 1000 into 1 this is simply meter cube per second this is kilograms per meter cube for water this is the specific heat which is kilocalories per kilogram degree C in the thermal pollution allowed is 1 degree C so if you multiply these out you get 2000 kilocalories per second this is the permitted thermal pollution if this is all the heat you can reject to the river then the work produced is automatically constrained by the formula that we already showed you have this is T1 this is T2 and this is T2 T2 actually increases by a degree does not make a difference here so you put this in and ask what is the work you can get this is joules conversion joules per calorie or kilo joules per kilo calorie so you get out of this you get power and you have to divide by 1000 to get megawatts this is joules per second because this is calories per kilo calories per second so if you like this is kilo joules per kilo calorie so if you convert this this is megawatts conversion to megawatts or if you like it is megawatt per kilowatt which is 1 by 1000 so you get 10.2 megawatts in the monsoon season the flow rate is 8 and your temperature T2 changes to 298.2 so you get the same figures it is slightly different 8000 instead of 2000 same formula the temperature T1 is the same T2 is now replaced 303.2 is replaced by 298.2 and then you have 4.18 the same conversion factors you get 41.2 megawatts so it is a fairly straight forward calculation we have seen 3 illustrative problems one is to follow the measurements of the masters to see what conclusions they came to the second we looked at problem of pumping water this is a normal day-to-day problem under in this case under adiabatic conditions the third problem that we looked at was simply the problem of power generation using both the laws what is the maximum power you can generate if you are constrained by the thermal pollution of the sink. So I hope these problems have illustrated for you the power of the thermodynamic theory that we did in class I will stop there in the next class we will discuss next session on tutorials we will discuss some more of the theory that we discuss in class thank you very much.